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Question about members in sets

Let $A_1,A_2,...,A_n$ be sets with $k$ members in $A_i$ for every $1\le i\le n$. Suppose that the $A_i$ satisfy: 1) $|A_i\cap A_j| = 1$ for all $i\ne j$, 2) $A_1\cap A_2\cdots\cap A_n =\emptyset$. ...